An object of mass m 1 hangs from a string that passes over a very light fixed pu

Question

An object of mass m1 hangs from a string that passes over a very light fixed pulley P1 as shown in the figure below. The string connects to a second very light pulley P2. A second string passes around this pulley with one end attached to a wall and the other to an object of mass m2 on a frictionless, horizontal table.

(a) If a1 and a2 are the accelerations of m1 and m2, respectively, what is the relation between these accelerations? (Use any variable or symbol stated above as necessary.)
a2 =

(b) Find expressions for the tensions in the strings in terms of the masses m1 and m2, and g.
T1 =
T2 =

(c) Find expressions for the accelerations a1 and a2 in terms of the masses m1 and m2, and g.
a1 =
a2 =

Explanation / Answer

a)

from the diagram, it is clear that for a given displacement of mass m1 in down direction , the displacement of m2 is twice in right direction

hence x2 = 2 x1

taking double derivative both side relative to "t"

a2 = 2 a1                         eq-1

b)

for block m2, force equation is given as

T2 = m2 a2 = 2 m2 a1

also

T1 = 2 T2

for hanging block m1, force equation is given as

m1 g - T1 = m1 a1

m1 g - T1 = T2 /2

but T1 = 2 T2

hence

m1 g - 2 T2 = T2 /2

m1 g = (2.5) T2

T2 = m1 g /2.5

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